Praise Pythagoras.

When a window family with arched lintel is necessary and height has to be controlled, what kind of parameters have you been using?

1. Angle based method

A arched lintel maintains it's angle if it has no parameters. It's just simple.

This can be efficient when proportion is a priority. And it's the easiest way.
But it makes difficult to predict total height of the window. It goes higher or lower depends on the width of the window.

2. Radius based method

Radius of lintel can be easily made to a parameter.

With this way, height of the lintel can be controlled separately by changing radius. But it's also hard to know peak of the lintel.

3. Height based method with Pythagorean Theorem.

It is well known that height of an arch can be found with Pythagorean Theorem.
I'll explain it again for someone who can't remember what teacher said in boring math class.

At the above image, we can get the formula with Pythagorean Theorem as follow.

radius^2 = (width/2)^2 + (radius-h)^2

so...  radius^2 = (width/2)^2 + radius^2 -2*radius*h + h^2
..........

so... radius = ((width / 2) ^ 2 + h ^ 2) / (2 * h)

This formula can find radius with given width and height of arch.

Now we can control the height of arched lintels like this.

Enjoy. :-)